Re: [xmca] Numbers - Natural or Real?

[Bill Kerr <billkerr@gmail.com>]

David,

I agree that play is important and your examples are good ones. My own
background in secondary teaching has incorporated a lot of play in maths
teaching using the ideas of Seymour Papert ("Mindstorms", logo), Idit Harel
(Children Designers) etc.

I am now teaching Australian aboriginal students of secondary school age but
primary school skills. eg. two students I taught last Thursday could not
halve  numbers beginning with an odd number, eg. 54. When I asked them how
many twos in five they said none. In the end to help them I had to draw 5
dots on a piece of paper and circle two lots of twos.

One thing I realised a little while back was that Papert, for all his great
work, did not provide much advice about numbers. Your initial suggestions
about playing with fractions initially and a game to follow up are helpful
in this regard.

I'd also be interested in the thoughts of anyone here about the Devlin
article.

On Sat, Jul 2, 2011 at 2:53 PM, David Kellogg <vaughndogblack@yahoo.com>wrote:

> Well, first of all, I wasn't responding to the Devlin article at all, Bill.
> I was responding to Andy's query about discourse and activity, Sfard and
> Davydov.
>
> I don't think that quantity IS the basic concept in mathematics, though.
> Vygotsky is pretty clear about this: just a preschooler has to be able to
> abstract actual objects away from groups in order to form the idea of
> abstract quantity, the schoolchild has to be able to abstract quantities
> away from numbers in order to form the idea of RELATIONS between quantities,
> or OPERATORS. That, and the idea of part-whole relations, are the
> cornerstones of algebra, which Vygotsky considers the true goal of math
> education in elementary school (he's very ambitious, and in some ways his
> programme makes more sense in a Korean context than in the West).
>
> We can easily see the difference when we consider how a given mathematical
> attitude considers the concept of INFINITY. If we consider infinity as a
> quantity, then there is no way to make sense of the statement that the
> infinity of whole numbers is larger than the infinity of even numbers (or
> the infinity of reals is larger than the infinity of naturals).
>
> But if we consider infinity as an OPERATION, that is, the recursive
> operation of adding to the last in a series, we get something rather
> different, which can explain this (given enough imagination on the part of
> the child).
>
> Bill, the reason I was pointing to PLAY as an important element in this was
> that I think the problem of teaching POTENTIAL as opposed to REAL is a
> recurrent problem in teaching. For example, when teachers want to teach "I
> can swim" in Korea they use a picture of a child swimming. But the result is
> that the children assume that "I can swim" essentially just means "swimming"
> or even "I am swimming" or "he is swimming".
>
> Play allows the teacher to solve this problem. In play, a real move is
> understood by the child as an instantiation of the potential moves within
> the imaginary situation or within the abstract rules. So the child easily
> grasps that an actual move is simply an instance of a potential.
>
> In the game I suggested, the main problem was showing the child how ANY
> number will do as a starting point, and how there is no ending point. Both
> of these are essential to teaching the point that each number is
> re-expressible in an infinite number of ways.
>
> But if the teacher suggests a number, the children will often take this
> number as the only number that will do. The teacher solves this problem by
> setting up a game in which the children suggest a number, and see that it
> can be any number.
>
> Similarly, if the teacher demonstrates a sequence (3/2, 6/4, 12/8 etc.) the
> children will see this as a complete or completable sequence. But if it is
> clear that the termination of the sequence is a LOSING move, then the idea
> of infinity as a process is within the child's grasp.
>
> David Kellogg
> Seoul National University of Education
>
> --- On Fri, 7/1/11, Andy Blunden <ablunden@mira.net> wrote:
>
>
> From: Andy Blunden <ablunden@mira.net>
> Subject: Re: [xmca] Numbers - Natural or Real?
> To: "Huw Lloyd" <huw.softdesigns@gmail.com>
> Cc: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>
> Date: Friday, July 1, 2011, 5:54 PM
>
>
> Well Huw, both the advocates of one or other method whose links I quoted
> believe there is a significant difference in choice of basic unit, and
> certainly from the point of view of the subject matter itself there is a
> difference: the concept of cardinal number is different from the concept of
> ordinal number, even though the non-mathematical adult probably never
> notices it. That distinction was one of the delightful insighsts I got from
> reading Anna's book. I knew the difference, but I just never reflected on it
> as something a child has to learn.
>
> I guess (apart from my question about foundations which I am hoping the
> experience of a maths teacher will shed light on) I am still working through
> my lived experience of discovering that the 14 year old kids I was teaching
> in 1975 who could add, subtract, multiply and even divide, had no concept of
> numbers as representing quantity, and never knew which arithmetical
> procedure to use in which practical situation outside of a small range of
> repeatedly rehearsed scenarios. That is, after 8 years of British public
> education, they had still not made the leap talked about in the Devlin
> article, which forms the beginning in Davydov's approach.
>
> Andy
>
> Huw Lloyd wrote:
>
>
>
> On 30 June 2011 09:52, Andy Blunden <ablunden@mira.net> wrote:
>
> I don't really have an opinion on this matter, but I would be interested in
> listening in on those who may have an informed opinion.
> I see two different approaches to the teaching of mathematics.
>
>   One takes the /Natural/ Numbers as the basic concept of the subject
>   The Other takes the /Rational/ Numbers as the basic concept of the
>   subject
>
>
> The Davydov example, described in "Cultural-Historical Approaches to
> Designing for Development" describes quantity as the basic concept.  Which
> seems sensible to me.
>
> Both magnitude and multitude are variants of quantity -- a concept that
> entails a number along with a unit of measure.
>
>
>   One takes /counting/ as the basic Action
>   The Other takes /comparison/ of two lengths as the basic Action.
>
>
> In both cases I would ascribe measuring as the "basic action", which
> includes the pattern matching found in counting apples.
>
> Huw
>
>
> --
>
>
> *Andy Blunden*
> Joint Editor MCA:
> http://www.informaworld.com/smpp/title~db=all~content=g932564744
> Home Page: http://home.mira.net/~andy/
> Book: http://www.brill.nl/default.aspx?partid=227&pid=34857
> MIA: http://www.marxists.org
>
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